Step by step
The question
Question: y varies inversely with x.
y = 12 when x = 5.
Find y when x = 8.
1 Write the rule
Varies inversely means y is a number divided by x.
When x gets bigger, y gets smaller.
We call the number k: y = k over x.
k stays the same for every pair of numbers.
Books call it the constant of variation.
2 Find k
We put in y = 12 and x = 5.
k/5 means k divided by 5.
We want just k.
To remove divided by 5, we multiply by 5.
We do it on both sides, so they stay equal.
Divided by 5, then times 5: they cancel.
Cross them out.
12 times 5 is 60.
So k = 60.
3 Write the equation
We put 60 in place of k.
So the equation is y = 60 over x.
4 Find y
We put in x = 8.
Careful!
Inverse is not direct.
x got bigger, from 5 to 8.
So y must get smaller.
We can make 60/8 simpler.
It is the same as 15/2.
So y = 15/2.
5 Check
Let's check.
In inverse variation, x times y is always k.
First pair: 5 times 12 is 60.
New pair: 8 times 15/2 is 60 too.
The same k.
So the answer is right!
Let's look back.
First: write the rule.
Then: find k.
Then: write the equation.
Then: find y.
Last: check.